If $E[X]<\infty$, X non-negativ then $X<\infty$ almost sure?

Question

Is it true that for an nonnegative randomvariable X we have $$E[X]<\infty \Rightarrow P(X<\infty)=1 \quad?$$

Answer 1

A slightly different example: consider Brownian motion with $B_t \sim N(0, t)$. Then, $\mathbf{E}[B_t]=0$. But, if you set event $E_n = \{B_n>|\varepsilon|\}$ for $n \in \mathbb{Z}$, then, using Borel-Cantelli lemma(II), $P(E_n \ i.o.) = \sum_{n=0}^{\infty}P(E_n) \to \infty$ because $P(E_n) >\Phi(-\frac{\varepsilon}{\sqrt{n}}) \to \frac{1}{2}>0$, hence the sum diverges, and $\limsup B_t = \infty$